D₁₅₇₄₅ #44538439

order	31490
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₇₄₅
unambiguous semidirect product	C₅⋊D₃₁₄₉
unambiguous semidirect product	C₄₇⋊D₃₃₅
unambiguous semidirect product	C₆₇⋊D₂₃₅

Splittings

Fitting, socle, derived, Hall	C₅×C₄₇×C₆₇ ⋊ C₂
Hall	C₄₇×C₆₇ ⋊ D₅
Hall	C₅×C₆₇ ⋊ D₄₇
Hall	C₅×C₄₇ ⋊ D₆₇
Sylow	C₆₇ ⋊ D₂₃₅
Sylow	C₄₇ ⋊ D₃₃₅
Sylow	C₅ ⋊ D₃₁₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₄₇×C₆₇	C₂
LCS	G ▷ C₅×C₄₇×C₆₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 15745

A minimal faithful permutation representation — 2 generators on 119 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(2,5)(3,4)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2,5,47,67|b^a=b^4,c^a=c^46,d^a=d^66

Conjugacy classes & character table

7874 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₇₄₅
maximal normal subgroup of	C₂×D₁₅₇₄₅
maximal quotient of	Dic₁₅₇₄₅, C₂×D₁₅₇₄₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31490, 8);

D15745 #44538439